====== Color sum ======

The **color sum** of a [[two-colored partition]] $p\in\Pscr^{\circ\bullet}$ is the integer-valued measure $\sigma_p:\mathfrak{P}(P_p)\to\Z$ on its set of points $P_p$ whose density assigns to any point $i\in P_p$ the value $1$ if $i$ is of [[normalized color]] $\circ$ in $p$ and $-1$ if $i$ is of normalized color $\bullet$ in $p$ (see [(:ref:MaWe19)], Section 3.3). 

Per definition, if $i$ is a lower point, then the normalized and the ordinary color of $i$ in $p$ coincide, and the two are opposites if $i$ is an upper point.

The color sum $\Sigma(p)\colon\hspace{-0.66em}=\sigma_p(P_p)$ of the set of all points of $p$ is called the **total color sum** of $p$ (see [(:ref:TaWe18)], Definition 2.4).

A set $S\subseteq P_p$ of points of $p$ is said to be **neutral** if its color sum vanishes, $\sigma_p(S)=0$ (see [(:ref:MaWe19)], Section 3.3).

====== References ======


[( :ref:TaWe18 >>
author:  Tarrago, Pierre and Weber, Moritz
title:   The classification of tensor categories of two-colored non-crossing partitions
year:    2018
journal: Journal of Combinatorial Theory, Series A
volume:  154
month:   February
pages:   464--506
url:     https://doi.org/10.1016/j.jcta.2017.09.003
archivePrefix: arXiv
eprint   :1509.00988 
)]

[( :ref:MaWe19 >>
author:  Mang, Alexander and Weber, Moritz
title:   Categories of two-colored pair partitions, part I: categories indexed by cyclic groups
year:    2019
journal: The Ramanujan Journal
url:     https://doi.org/10.1007/s11139-019-00149-w
archivePrefix: arXiv
eprint   :1809.06948
)]